A horsepond is a small body of water for the watering of horses, or a pond about the size that would serve that function. In sailing terms, this is a label denoting insignificance.
Molland's was a shop on Milsom Street in Bath where confections could be purchased and enjoyed. Anne and her party briefly stop at the shop to avoid the rain.
To this day, Milsom Street remains the main shopping street in Bath.
A type of carriage, the barouche was a stately horse-drawn vehicle with a top that could be taken down or put up depending on the weather or moods of the occupants.
Typically seating four people, on two seats facing each other, a barouche like Lady Dalrymple's would have been an essential form of fashionable and elegant transportation.
Captain Wentworth shows concern for Anne Elliot's umbrellaless condition on a rainy Bath day by encouraging her to use a sedan chair as a means of returning home. Sedan chairs were, as illustrated, a human-powered vehicle composed of an enclosed box and two poles. Decoration varied (this illustration is of a rather elaborate design for Queen Charlotte).
The poles would be manned, quite literally, and the chair's occupant would then be carried to his or her destination in relative comfort, if not speed.
The theatre is most likely the Theatre Royal and the rooms referred to here are the Assembly rooms, of which the octagon room (below) is one. The former was for plays and theatricals, the latter for balls, parties, and social elbow-rubbing.
The vanity and social ambition of Miss Elliot and Sir Walter are again revealed in their disdain of the forms of entertainment which are good enough for gentlemen like Captain Wentworth and the Crofts.
Both are still lively venues today. For more about the Theatre Royal, try their official site.
The octagon room was one of the assembly (or Upper) rooms for society and entertainment in Regency Bath.
The room from which the other assembly rooms branched out, it was often used for the same purpose -- a concert -- that brought Anne Elliot and her party together in this context. These rooms are still used for similar purposes today.